The Art of the Variance Swap
Dean Curnutt, senior salesperson in the capital markets group at Commerzbank, explains how portfolio managers can use variance swaps to manage volatility risk.
Derivatives market professionals know that managing volatility is central to hedging the risk in an options portfolio. Although the BlackScholesMerton framework for hedging options is both wellestablished and wellunderstood, spectacular losses in volatility trading have been dealt to brokerdealers and hedge funds in the past.
Alas, the tools for managing volatility risk are few. But a relatively new product—the variance swap—offers investors a straightforward vehicle for achieving long or short exposure to market volatility. Although it's called a swap contract, it is fundamentally an optionbased product with properties similar to those of options. The product consequently represents a significant addition to the overall landscape of volatilitydriven instruments and can fill a useful role for investors seeking optionality in one form or another.
How it works
The variance swap is a contract in which two parties agree to exchange cash flows based on the measured variance of a specified underlying asset during a certain time period. On the trade date, the two parties agree on the strike price of the contract (the reference level against which cash flows are exchanged), as well as the number of units in the transaction.
For example, suppose two parties agreed to trade a sixmonth variance swap on the Standard & Poor's 500 index with a strike price of 25 percent and a unit amount of 50,000. Suppose realized standard deviation of the S&P 500 during this time period turns out to be 30 percent. The payoff to the party that receives volatility is 50,000 x (0.302 – 0.252) x 100, or 137,500. If realized standard deviation were 20 percent instead, the payoff to the party that pays volatility would be 50,000 x (0.252 – 0.202) x 100, or 112,500. Figure 1 illustrates the payoff of a short variance swap under different levels of realized volatility.
The above example highlights an important property of the variance swap: Its payoff is nonlinear in volatility. This means, for instance, that a 1 percent deviation of realized volatility above the strike price has a different (larger) payoff than a 1 percent deviation of volatility below the strike price. These differences are generally insignificant for small deviations from the strike price, but can be large when realized volatility is materially different from the strike price.
Pricing and hedging
The economic characteristics of the variance swap are similar to those of an option contract. Like an option, the value of a variance swap is influenced by both realized and implied volatility, as well as the passage of time. A portfolio consisting of an appropriately weighted combination of option contracts across different strikes can be constructed to hedge a variance swap. In general, such a hedge would be designed to render the vega exposure constant across different strikes. By weighting the number of options according to the inverse of the strike squared, a constant vega profile can be achieved and would effectively hedge the variance swap.
The payoff of a short variance swap with a strike price of 25 percent under different levels of realized volatility.
The P/L of a deltaneutral option position maintained until expiration reflects the aggregate gamma capture from hedge rebalancing netted against the premium paid for the option. Since gamma itself is a complex function of time, volatility and the option's "moneyness” (the relationship of the stock price to the strike price), this P/L is subject to inherent uncertainties. The P/L of a variance swap, on the other hand, is a direct function of realized volatility vs. the swap strike level, and is thus significantly easier to describe.
Pricing a variance swap is an exercise in computing the weighted average of the implied volatilities of the options required to hedge the swap. That is, the strike price is set so as to reflect the aggregate cost (in implied volatility terms) of the hedge portfolio. To see this more clearly, suppose the strike price on the variance swap were set to zero. Since standard deviation (and thus variance) is always nonnegative, the payoff of this contract would always be positive and the contract would necessarily carry a cost. This differs from market convention in which variance swaps are entered into without any initial cash flow. The cost of the zerostrike variance swap, then, is simply the sum of the option premium expended in purchasing the correct hedge portfolio.
Applications
One of the most significant applications for variance swaps is in the area of volatility trading. For investors who have traditionally employed deltaneutral option strategies to implement views on volatility, the variance swap offers a more exact method for taking views on future volatility. Deltaneutral long option strategies are based on buying options that carry an implied volatility that is less than the volatility that will ultimately be realized. Conversely, short option strategies are based on selling options at an implied volatility that is rich compared with the anticipated realized volatility. The profitability of these strategies, however, depends on the complicated interaction of factors such as movements in the underlying asset and the passage of time. Variance swaps provide a cleaner way of speculating on realized vs. implied volatility.
 Without Variance Swap;  With Variance Swap
Some of the possible strategies using variance swaps include the following:
 Speculating that implied volatility is too high or too low relative to anticipated realized volatility.
 Implementing a view that the implied volatility in one equity index is mispriced relative to the implied volatility in another equity index.
 Trading volatility on a forward basis by purchasing a variance swap of one expiration and selling a variance swap of another expiration.
In a variance swap, two parties agree to exchange cash flows based on the measured variance of a specified underlying asset during a certain period. 
These strategies are initiated by the investor in search of profit—that is, they are proactive in nature. Nonetheless, the variance swap is also potentially useful as a defensive strategy that seeks to protect a portfolio from a market selloff. Since a significant market decline is usually accompanied by an increase in volatility, a long variance swap position will help offset portfolio losses that result from a steep market decline. In falling markets, volatility typically rises—on a realized basis and on an implied basis, since hedging short gamma positions becomes riskier. Figure 2 shows the effectiveness of a variance swap in hedging a portfolio indexed to the S&P 500 during the turbulent month of August 1998. (The graph assumes that the investor owns $10 million worth of the S&P 500 index and implements a variance swap with a strike of 25 percent on 100,000 units.) The graph illustrates the daily fluctuation of this portfolio with and without the variance swap.
With broad equity indices trading at record P/E multiples, many in the investment community believe that a bubble in financial asset prices has developed and that valuations can't stretch much further. Others argue that a new paradigm has arrived and that traditional metrics for assessing stock price valuation no longer apply. Whatever the case, it's likely the divergence of opinion will continue to produce an environment of uncertainty. And uncertainty leads to volatility. The variance swap therefore represents an important new product for managing volatility risk and taking proactive views in a market in which substantial daytoday price fluctuations are increasingly the norm.
Quantifying Volatility Convexity
DerivaTech's Howard Savery explains how veganeutral butterflies and zeta can be used to address volatility convexity in option pricing.
The BlackScholesMerton option pricing framework provides market practitioners with a simple and effective model for valuing Europeanstyle vanilla options (EV). The acceptance of this model as the standard tool for option pricing was the crucial ingredient for the growth of an efficient market through the quoting of EV options by implied volatility. The model's assumption that volatility and deposit rates are constant through different underlying points and time points leaves only the spot price to exhibit stochastic properties in its Brownian motion (lognormal distribution of rates of return). The model's valuation of an EV option can therefore be attributed to the convexity of its price to changes in the underlying price. In other words, the EV call (put) will increase at an increasing rate as the underlying rises (falls), and will decrease at a decreasing rate as the underlying falls (rises). The value is therefore convex to changes in the underlying price.
The BlackScholesMerton model, however, ignores another extremely important convexity measurement: volatility convexity. The price of an EV option increases at an increasing rate as volatility rises, and decreases at a decreasing rate as volatility falls. In essence, this is additional optionality, which the BlackScholesMerton formula misses. We'll now examine a method to quantify the cost of convexity to volatility changes.
The trading community overcame the BlackScholesMerton assumption of a constant volatility by adjusting the implied volatility for each strike so that the resulting BlackScholesMerton prices reflect expectations of volatility variations. The result is the volatility smile with implied volatility increasing as strike prices move away from the atthemoney (ATM) price. For the purposes of this discussion, we will assume an ATM volatility of 10 percent with a typical smile curve.
The BlackScholesMerton model ignores an extremely important convexity measurement: volatility convexity. 
ATM EV options do not exhibit convexity to volatility. That is, the vega (change in price for change in volatility) for ATM EV options is essentially constant for changes in volatility; thus the change in price for change in volatility is linear. Outofthemoney (OTM) EV options, on the other hand, do exhibit volatility convexity. They have a vega that increases as volatility rises, and decreases as volatility falls. Consequently, the change in price for a change in volatility is convex. Because of this convexity, OTM EV options are quoted at a volatility that is greater than the volatility for ATM EV options (in the absence of skew). The difference in volatility is an adjustment made to account for vega convexity under the BlackScholesMerton framework.
We can use a structure known as a veganeutral butterfly to quantify the cost of convexity. A veganeutral butterfly is defined as the sale of an ATM straddle combined with the purchase of an OTM strangle. The straddle and strangle amounts are set at a ratio that leaves the position initially veganeutral. The buyer of the butterfly would have a position that is initially veganeutral but that has positive convexity to volatility. The average change in the butterfly vega over unit changes in the implied volatility would then be measured. This concept of measuring vega convexity by the average change in vega is known as dvegadvol.
Now that there is a measure of convexity, the implied cost of this convexity needs to be measured. Zeta, the difference between the value of an option using the ATM volatility (BlackScholesMerton) and its value at the marketadjusted volatility will be our measure of the cost. Figure 1 shows the vega of the butterfly plotted for three different levels of volatility as the spot price changes.
With the increased popularity of various exotic options, the risk of model error from assuming constant volatility has become magnified. 
With the measures of butterfly convexity and zeta, we can calculate the implied unit cost of vega convexity as zeta divided by dvegadvol. For example, with a convexity (dvegadvol) of 0.01275 and a zeta of 0.08, we can derive a cost of convexity of $6.275. To put this in market terms, a long veganeutral butterfly with a dvegadvol of $1,000 for a 1 percent change in volatility would cost $6,275 more than the value calculated at 10 percent flat volatility. We can then use this measure to compare the convexity and cost of other option structures by applying the unit cost of convexity and calculating a fair value of the zeta that is consistent with the convexity.
With the increased popularity of various exotic options, the risk of model error from assuming constant volatility has become magnified. Most exotic options are at least initially modeled in a BlackScholesMerton framework, which assumes constant volatility. While the method of adjusting the implied volatility to take volatility convexity into account was sufficient for EV options, the same methods cannot be used for many exotics. The most common misunderstanding in pricing exotics comes from a misconception of what the implied volatility smile actually means. If the implied volatility quoted for OTM EV options is a volatilityconvexityadjusted parameter used to overcome the shortcoming of the BlackScholesMerton formula, then simply to apply this volatility to exotic options with a quite different convexity is inappropriate. What is needed is a method to infer the cost of vega convexity and then to apply this cost on a relative convexity basis to all products.
Howard Savery, senior vice president for business development at DerivaTech Consulting LLC, can be reached at howie@derivatech.com.
